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1.1 Decomposition group and inertia group. ... more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local ...
The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ switches the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
For example, if L is a Galois extension of a number field K, the ring of integers O L of L is a Galois module over O K for the Galois group of L/K (see Hilbert–Speiser theorem). If K is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of K and its study leads to local class field theory.
In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups. In this article, a local field is non-archimedean and has finite residue field.
In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) wild (non-tame) ramification. This goes beyond the geometric analogue.
The ramification defect or ramification deficiency d of a valuation of a field K is given by [L:K]=defg where e is the ramification index, f is the inertia degree, and g is the number of extensions of the valuation to a larger field L. The number d is a power p δ of the characteristic p, and sometimes δ rather than d is called the ...
Decomposition group; Inertia group; Frobenius automorphism; Chebotarev's density theorem; Totally real field; Local field. p-adic number; p-adic analysis; Adele ring; Idele group; Idele class group; Adelic algebraic group; Global field; Hasse principle. Hasse–Minkowski theorem; Galois module; Galois cohomology. Brauer group
The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : I m → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m.